Would I be able to add an Airport Express into the dining room to extend the Extreme's network and just plug the NAS into the Express?
Do you mean in place of the dining room switch...? If so, then it's no different - the "LAN" ports in a SOHO router are a built-in switch so offer no improvement over a stand alone switch.
If you want to keep the Extreme as your routing device, then whatever devices you require on your internal network have to be downstream of it's WAN/LAN boundary.
So you could do something like this...
ISP+++[WAN]SH(modem mode)[LAN]-----[WAN]Airport(configured as router/AP.)[LAN]----switch/NAS/et/al
....but that would mean moving the Extreme into you dining room (if I'm reading your geography right) with the NAS which may clobber wi-fi. Thence to fix that you'd have to add another wi-fi device in your lounge.
Alternatively the same topology could be acheived if you could pull an extra network cable between lounge and dining room. You may then not need the switch if the only dining room device is your NAS.